Balanced and Strongly Balanced Bull-Designs

An H-system of order v is a partition of the edge-set of the complete graph Kv such that each element of the partition induces a subgraph isomorphic to the graph H and the graphs of the partition are said to be the blocks. An H-system is said to be balanced if the number of blocks containing any given vertex of is a constant. An H-system is called strongly balanced if for every i =1,2, ... ,h, there exists a constant such that (x) = for every vertex x, where s are the orbits of the automorphism group of H on its vertex-set and (x) is the number of blocks of containing x as an element of . We say that an H-system is simply balanced if it is balanced, but not strongly balanced. In this paper, we show that (i) a strongly balanced bull-design of order v exists if and only if v ≡ 1 (mod 10) and (ii) a simply balanced bull-design of order v exists if and only if v ≡ 1 or 5 (mod 10), v ≠ 11.